A cylindrical buoy 1.8m in diameter, 1.2m high and weighing 10.5 KN floats in salt water of density 1025 kg/m3. Its CG is 0.45m from the bottom. If a load of 3KN is placed on the top, find the maximum height of the CG of this load above the bottom if the buoy is to remain in stable equilibrium.

Problem Statement

A cylindrical buoy with the following properties floats in salt water:

Diameter: 1.8m
Height: 1.2m
Weight: 10.5 kN
CG of buoy from bottom: 0.45m
Salt water density: 1025 kg/m³

A 3kN load is placed on top of the buoy. Determine the maximum height of the CG of this load above the bottom for stable equilibrium.

Solution

1. Calculate the Depth of Immersion (\(h\))

The total weight of the buoy and load equals the weight of the displaced salt water: \[ 10500 + 3000 = \rho_{\text{salt water}} g V_{\text{displaced water}} \] \[ 13500 = 1025 \times 9.81 \times \pi \times (0.9)^2 \times h \] \[ h = 0.53m \]

2. Compute the Center of Buoyancy (\(OB\))

The center of buoyancy is at the centroid of the submerged volume: \[ OB = \frac{h}{2} = \frac{0.53}{2} \] \[ OB = 0.265m \]

3. Compute the Metacentric Radius (\(MB\))

\[ MB = \frac{I}{V} \] The moment of inertia about the vertical axis: \[ I = \frac{1}{64} \pi (1.8)^4 \] The displaced volume: \[ V = \frac{\pi}{4} (1.8)^2 \times 0.53 \] \[ MB = 0.38m \]

4. Compute the New Center of Gravity (\(G’\))

\[ BG’ = OG’ – OB \] \[ G’M = MB – BG’ \] \[ G’M = 0.38 – OG’ + 0.265 = 0.645 – OG’ \] For stable equilibrium: \[ G’M > 0 \] \[ 0.645 – OG’ > 0 \] \[ OG’ < 0.645m \]

5. Compute the Maximum Height of Load CG (\(OG_1\))

Taking moments about \(O\): \[ 3 \times 10^3 \times OG_1 + 10.5 \times 10^3 \times 0.45 = (3 \times 10^3 + 10.5 \times 10^3) \times 0.645 \] \[ OG_1 = \frac{(13.5 \times 0.645) – (10.5 \times 0.45)}{3} \] \[ OG_1 = 1.3275m \]

Final Result:

Maximum height of CG of load above the bottom: 1.3275m

Explanation

1. Stability Condition:
A floating body is stable if its metacentric height (\(GM\)) is positive. This means the restoring moment is greater than the overturning moment.

2. Calculation of CG Shift:
– When a load is placed on top, the center of gravity (\(G’\)) shifts upwards.
– The stability condition ensures that the new CG does not exceed the critical height for stability.

3. Importance of Stability Analysis:
– Ensures floating structures do not tip over under additional loads.
– Used in marine engineering, ship design, and offshore platforms.

Physical Meaning

1. Engineering Applications:
– Used in designing floating buoys and marine structures.
– Helps in weight distribution analysis in floating platforms.

2. Industrial and Real-World Uses:
– Used for designing oil rigs, submarines, and offshore energy platforms.
– Helps in determining safe loading conditions for ships and floating docks.

A cylindrical buoy 1.8m in diameter, 1.2m high and weighing 10.5 KN floats in salt water of density 1025 kg/m3. Its CG is 0.45m from the bottom. If a load of 3KN is placed on the top, find the maximum height of the CG of this load above the bottom if the buoy is to remain in stable equilibrium.